Error‐systematics of determining simultaneously the isotopic abundance ratios of natural lithium and natural boron as Li ₂ BO ₂ ⁺

Abstract

Recently, we have shown how the errors delta(j) and delta(k), that occur when measuring the two different isotopic molecular abundance ratios required for analysis, are transformed into the actual errors of elemental isotopic analysis, (deltaEi/Ei)'s. With a view to gain further understanding as to how the errors (deltaEi/Ei)'s are governed, we now evaluate theoretically the effects of selecting different isotopic molecular pairs as the monitor pairs (j and k) for measurements, and of the measurement errors (delta(j) and delta(k)), on the results of analysis (the 6Li/7Li and the 10B/11B abundance ratios), by considering all the constituent elements of Li2BO2+ at their natural isotopic abundances. It is shown that the ratio of measurement errors, delta(j)/delta(k), is a more fundamental parameter than either the individual errors (delta(j) and delta(k)), or their sum, absolute value(delta(j)) + absolute value(delta(k)), in governing deltaEi/Ei. The important implication of this observation is that it reveals the possibility of achieving not only a desired level of accuracy in analysis, but even absolute accuracy (i.e. deltaEi/Ei = 0) by causing mutual cancellation of the effects of individual measurement errors delta(j) and delta(k), through proper regulation of measurement parameters. However, as the measurement errors cannot be pre-set, it is shown how selection of proper monitor pairs (j and k) can help achieve the desired accuracy in analysis. The present work sets guidelines for the more general problem of selecting monitor pairs to avoid larger errors in analysis.

Error-systematics of determining simultaneously the6Li /7Li and the10B/11B abundance ratios as Li2BO2+: considerations for analyte elements of varying isotopic abundance patterns

“…However, for given δ j and δ k , the |δ E > can be obtained, as has already been shown elsewhere,2,, 3 as follows: where the notation |δ x | indicates the absolute value of δ x , and the notation |X i > denotes the column vector whose elements are the X i (where X denotes a general variable). The parameter |[REMF] E jk >, which depends2,, 3 obviously on the relationships between E i 's and R i 's (Eqns 1 and 2), is referred to generally as the relative error magnification factor. However, it is defined2,, 3 to be (numerically) equal to the ratio of actual error in analysis to the measurement errors, as follows: …”

“…The parameter |[REMF] E jk >, which depends2,, 3 obviously on the relationships between E i 's and R i 's (Eqns 1 and 2), is referred to generally as the relative error magnification factor. However, it is defined2,, 3 to be (numerically) equal to the ratio of actual error in analysis to the measurement errors, as follows: …”

“…However, it was observed2,, 3 that |[REMF] E jk > is not purely a theoretical parameter decided by the functions, f j (L, Y) and f k (L, Y) [Eqns 1 and 2], but depends on the experimental errors (δ j and δ k ) also. Actually, it has been shown2,, 3 that, even for a specified analyte isotopic ratio (E i ) and a chosen CTMPS as the monitor pairs (j and k), the |[REMF] E jk > will be fixed if only two other different kinds of parameters, namely, the molecular IAD and the measurement errors (δ j and δ k ), are also fixed.…”

“…Thus, for the case in which all three constituent elements of Li 2 BO 2 + are of natural origin, it was demonstrated3 how, for any given measurement errors (δ j and δ k ), the |[REMF] E jk > and hence the accuracy of analysis (|δ E >) vary with the choice of monitor pairs (j and k). Similarly, for chosen monitor pairs, it was also clarified3 how |[REMF] E jk > is related to δ j and δ k . It was shown3 that |[REMF] E jk > is, however, independent of the actual magnitudes of δ j and δ k , and is controlled only by their ratio, δ j /δ k .…”

“…Similarly, for chosen monitor pairs, it was also clarified3 how |[REMF] E jk > is related to δ j and δ k . It was shown3 that |[REMF] E jk > is, however, independent of the actual magnitudes of δ j and δ k , and is controlled only by their ratio, δ j /δ k . This observation thus led3 to the possibility that, irrespective of the CTMPS chosen as monitor pairs (j and k), proper regulation of the measurement parameters would help achieve not only (|[REMF] E jk >)<1, i.e.…”